I just bought an iPad Nano 7th generation, after using a Cowon player for a few years (got tired of Apple's subpar SQ). Since my music collection is an ALAC iTunes libary, an iPod is the most practical choice. Now, Cowon has great EQ and SFX options, but Apple STILL only offers a few crappy sounding and unusable EQ presets. The user cannot tweak and make his/her own settings. This is getting downright ridiculous. You can buy an iPod with a camera, games, play videos etc., but having absolute basic audio functions like a decent tweakable EQ...oh no. Not to mention that all the EQ presets are way to exaggerated, and are never going to be used by anyone with a slightly musical ear. EQ or similar SFXs are very useful for certain types of headphones, esp. IEMs (inner ear monitors)
Somehow I'm not that suprised...

I have an 6th gen, and the SQ is very good (way better than the dead classic that I still have lying around).

About the equalizer Apple never really had a decent one. Even Sony had a simple, yet kinda decent equalizer. But Apple have improved (slightly) the eq, on the classic activating any preset that increased the output of some bands, like the bass boost one, would make music start clipping.

Either get some headphones that don't need eq, some processing unit of some kind or simply deal with it.

You can apply an EQ in iTunes and it will sync over. You have to apply it specifically to the song though.

I did try this a couple of years ago, and couldn't get it to work then. It is a very unpractical solution anyway. Yes, the SQ has been slightly improved in the iPod Nanos the few last years, but again Apple is downplaying the core function of the iPod - music listening.